The main characters in this paper are the objects \(A_u(Q)\)  mentioned in the introduction. Here, \(Q\)  is a positive invertible \(n\times n\)  matrix, and by definition, \(A=A_u(Q)\)  is the \(*\) -algebra freely generated by \(n^2\)  elements \(u_{ij}\) , \(i,j=\overline{1,n}\)  subject to the conditions that both \(u=(u_{ij})_{i,j}\)  and \(Q^{\frac{1}{2}}\overline{u}Q^{-\frac{1}{2}}\)  be unitary \(n\times n\)  matrices in \(A\)  (where \(\overline{u}=(u_{ij}^*)_{i,j}\) ); cf. [1]}, or [2]}, where these objects were first introduced in their \(C^*\) -algebraic versions.
